The output y[n] of a discrete-time system for an input x[n] is

$y\left[n\right]=\underset{-\mathrm{\infty }\le k\le n}{max}\left|x\left[k\right]\right|$

The unit impulse response of the system is

Concept:

The unit impulse response of a system is the response when the input is a unit impulse.

This is explained as:

y(n) = x(n) ⊕ h(n)

For a unit impulse input, the output will be:

y(n) = h (n)

Calculation:

Given:

$y\left[n\right]=\underset{-\mathrm{\infty }\le k\le n}{max}\left|x\left[k\right]\right|$

To find the impulse response of a system, the input will be an impulse, i.e.

x(k) = δ(k)

So,$y\left[n\right]=\underset{-\mathrm{\infty }\le k\le n}{max}\left|\delta \left[k\right]\right|$

For n < 0:

$y\left[n\right]=\underset{-\mathrm{\infty }\le k\le n}{max}\left|\delta \left[k\right]\right|=0$

For n = 0:

y[n] = δ(0) = 1

For n > 0:

$y\left[n\right]=\underset{-\mathrm{\infty }\le k\le n}{max}\left|\delta \left[k\right]\right|=1$

$\therefore y\left(n\right)=\left[\begin{array}{cc}0;& -\mathrm{\infty }<n<0\\ 1;& 0\le n<\mathrm{\infty }\end{array}\right]$